Fuzzy Graphs

In this paper, neighbourly irregular fuzzy graphs, neighbourly total irregular fuzzy graphs, highly irregular fuzzy graphs and highly total irregular fuzzy graphs are introduced. A necessary and sufficient condition under which neighbourly irregular and highly irregular fuzzy graphs are equivalent is provided. We define d2 degree of a vertex in fuzzy graphs and total d2 -degree of a vertex in fuzzy graphs and (2, k)-regular fuzzy graphs, totally (2, k)- regular fuzzy graphs are introduced. (2, k)- regular fuzzy graphs and totally (2, k)-regular fuzzy graphs are compared through various examples

A study on irregularity in vague graphs with application in social relations

Considering all physical, biological and social systems, fuzzy graph models serves the elemental processes of all natural and artificial structures. As the indeterminate information is an essential real-life problems, which are mostly uncertain, modelling those problems based on fuzzy graph is highly demanding for an expert. Vague graph can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems, in which fuzzy graphs possibly will not succeed into bringing about satisfactory results. Also, vague graphs are so useful tool to examine many issues such as networking, social systems, geometry, biology, clustering, and traffic plan. Hence, in this paper, we introduce strongly edge irregular vague graphs and strongly edge totally irregular vague graphs. A comparative study between strongly edge irregular vague graphs and strongly edge totally irregular vague graphs is done. Finally, we represent an applicationof irregular vague influence graph to show the importance of irregularity in vague graphs.Publisher's Versio

Distance Metric Learning using Graph Convolutional Networks: Application to Functional Brain Networks

Evaluating similarity between graphs is of major importance in several computer vision and pattern recognition problems, where graph representations are often used to model objects or interactions between elements. The choice of a distance or similarity metric is, however, not trivial and can be highly dependent on the application at hand. In this work, we propose a novel metric learning method to evaluate distance between graphs that leverages the power of convolutional neural networks, while exploiting concepts from spectral graph theory to allow these operations on irregular graphs. We demonstrate the potential of our method in the field of connectomics, where neuronal pathways or functional connections between brain regions are commonly modelled as graphs. In this problem, the definition of an appropriate graph similarity function is critical to unveil patterns of disruptions associated with certain brain disorders. Experimental results on the ABIDE dataset show that our method can learn a graph similarity metric tailored for a clinical application, improving the performance of a simple k-nn classifier by 11.9% compared to a traditional distance metric.Comment: International Conference on Medical Image Computing and Computer-Assisted Interventions (MICCAI) 201

Sparse random graphs: regularization and concentration of the Laplacian

We study random graphs with possibly different edge probabilities in the challenging sparse regime of bounded expected degrees. Unlike in the dense case, neither the graph adjacency matrix nor its Laplacian concentrate around their expectations due to the highly irregular distribution of node degrees. It has been empirically observed that simply adding a constant of order $1/n$ to each entry of the adjacency matrix substantially improves the behavior of Laplacian. Here we prove that this regularization indeed forces Laplacian to concentrate even in sparse graphs. As an immediate consequence in network analysis, we establish the validity of one of the simplest and fastest approaches to community detection -- regularized spectral clustering, under the stochastic block model. Our proof of concentration of regularized Laplacian is based on Grothendieck's inequality and factorization, combined with paving arguments.Comment: Added reference

High performance subgraph mining in molecular compounds

Structured data represented in the form of graphs arises in several fields of the science and the growing amount of available data makes distributed graph mining techniques particularly relevant. In this paper, we present a distributed approach to the frequent subgraph mining problem to discover interesting patterns in molecular compounds. The problem is characterized by a highly irregular search tree, whereby no reliable workload prediction is available. We describe the three main aspects of the proposed distributed algorithm, namely a dynamic partitioning of the search space, a distribution process based on a peer-to-peer communication framework, and a novel receiver-initiated, load balancing algorithm. The effectiveness of the distributed method has been evaluated on the well-known National Cancer Institute’s HIV-screening dataset, where the approach attains close-to linear speedup in a network of workstations

Generalized stepwise transmission irregular graphs

The transmission ${\rm Tr}_G(u)$ of a vertex $u$ of a connected graph $G$ is the sum of distances from $u$ to all other vertices. $G$ is a stepwise transmission irregular (STI) graph if $|{\rm Tr}_G(u) - {\rm Tr}_G(v)|= 1$ holds for any edge $uv\in E(G)$. In this paper, generalized STI graphs are introduced as the graphs $G$ such that for some $k\ge 1$ we have $|{\rm Tr}_G(u) - {\rm Tr}_G(v)|= k$ for any edge $uv$ of $G$. It is proved that generalized STI graphs are bipartite and that as soon as the minimum degree is at least $2$, they are 2-edge connected. Among the trees, the only generalized STI graphs are stars. The diameter of STI graphs is bounded and extremal cases discussed. The Cartesian product operation is used to obtain highly connected generalized STI graphs. Several families of generalized STI graphs are constructed